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A330138 List of pairs [j,k] ordered by increasing j and k according to the procedure described in the Comments. 0
3, 4, 5, 15, 8, 17, 5, 12, 13, 35, 12, 37, 21, 20, 29, 7, 24, 25, 63, 16, 65, 45, 28, 53, 27, 36, 45, 9, 40, 41, 99, 20, 101, 77, 36, 85, 55, 48, 73, 33, 56, 65, 11, 60, 61, 143, 24, 145, 117, 44, 125, 91, 60, 109, 65, 72, 97, 39, 80, 89, 13, 84, 85 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Procedure: The sequence lists two indices, j and k. Let (a, b, and c) represent the two legs and the hypotenuse of a right-angled triangle. The required computations are as follows:

(1) compute  2*j^2    (call this p, it is c-a)

(2) compute (2*k-1)^2 (call this q, it is c-b)

(3) compute 2j*(2k-1) (call this r, it is a+b-c)

So c = p+q+r; a = c-p; b = c-q.

Most, but not all, of the Pythagorean triples generated here are primitive (i.e., they have sides that are relatively prime). In the first 105 Pythagorean triples, there are 14 that are not primitive. The first exception is the 9th in the list: 27,36,45, where j=3 and k=2.

LINKS

Table of n, a(n) for n=1..63.

Wikipedia, Dickson's method of generating Pythagorean triples.

EXAMPLE

As the values of j and k increase, so do the lengths of the sides of the triangles. So it makes sense to order the triangles based on the sum (j+k).  When j=k=1, the sum (j+k)=2, and there is only one triangle (3,4,5).  When the sum of j and k is 3, there are two possibilities: j=1, k=2 (15,8,17) or j=2, k=1 (5,12,13).  When the sum is 4, there are three possibilities. Each time the sum (j+k) increases by one, the number of triangles also increases by one. And the total number of triangles grows according to sequence A000217 (the triangular numbers).

Value of j+k:    2, 3, 4,  5,  6,  7,  8,  9, 10, 11, 12, ...

Total triangles: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...

So it is clear that triangle number 61 would be in the range of triangles ranked 56 through 66, where (j+k)=12.  In fact, exactly 6 beyond the triangle associated with the values j=10, k=1.  So j=6, k=6 is associated with the sixty-first triangle. And the computations give us this Pythagorean triple: a=253, b=204, c=325.

MATHEMATICA

Table[Map[{#2 + #3, #1 + #3, #1 + #2 + #3} & @@ {2 #1^2, (2 #2 - 1)^2, 2 #1 (2 #2 - 1)} & @@ {#, n - #} &, Range[n - 1]], {n, 7}] // Flatten (* Michael De Vlieger, Dec 14 2019 *)

CROSSREFS

Cf. A103606.

Sequence in context: A049895 A226117 A221173 * A051530 A240670 A048040

Adjacent sequences:  A330135 A330136 A330137 * A330139 A330140 A330141

KEYWORD

nonn,tabf

AUTHOR

Joseph Damico, Dec 02 2019

STATUS

approved

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Last modified April 5 10:28 EDT 2020. Contains 333239 sequences. (Running on oeis4.)